A braid apart
Mathematics is often viewed as a male domain, but the field called symplectic geometry seems to be attracting an inordinate number of women. Among its leaders is University of Toronto professor Lisa Jeffrey, who weaves together geometry and quantum mechanics to provide proof for some far-out physics theories.
Mathematician Lisa Jeffrey is a bright splash of colour and vibrant energy on a dull and dreary afternoon in Toronto. Dressed in a neat red suit with a fashionably tied scarf at the throat, she sits forward in her chair in the sitting area of the Fields Institute like an intensely focussed hummingbird. Dr. Jeffrey, at age 40, is a world leader in the complicated mathematics of symplectic geometry and one of a notable cluster of women in an area of physics-linked mathematics that has blossomed over the last two decades.
"There is a clustering of women in symplectic geometry," says Dr. Jeffrey, a professor at the University of Toronto at Scarborough. "Everyone comments on it."
This is the woman who kicked her career into gear by building the mathematical foundation and structural supports for the brilliant but abstract 11-dimensional ideas floated by string theorist Ed Witten, the resident genius at the Institute for Advanced Study in Princeton, New Jersey. Dr. Jeffrey did her post-doc with Dr. Witten, giving her lots of time to absorb his arguments and develop some insight into his results. She teamed up with a colleague, Frances Kirwan from Oxford University, to work on mathematical proofs to support Dr. Witten's work.
"We were able to take this much farther than I expected and published three papers, proving Witten's results," says Dr. Jeffrey. She was always quite confident, she adds, that Dr. Witten's high-flown ideas would be amenable to mathematical proofs.
But the proofs that the two younger mathematicians established are just the tip of the iceberg. Dr. Witten is sometimes considered a latter-day Isaac Newton, in that both men are known for taking inspired leaps to solve physics problems without worrying about mathematical proofs to support their work. Newton invented calculus to provide the math behind classical physics (although, depending on your reading of history, Gottfried Leibniz did the same thing at the same time), but it took more than two centuries for mathematicians to come up with a theory to explain Newton's methods. In the same way, Dr. Witten's prodigious work on string theory - the idea that the universe is constructed of very, very tiny vibrating loops or strings rather than very, very tiny particles - is providing challenges to mathematicians today and driving new developments in math.
Many of these challenges were under discussion at the recent Strings05 conference, hosted this time by the University of Toronto. The annual conference is the major international gathering for string theorists. Dr. Witten was in attendance and was reportedly treated with awe by many participants.
String theory, say its adherents, essentially picks up where Einstein left off in his search for the so-called general unified theory, which tried to unite all the forces of nature. String theory seeks to reconcile Einstein's theory of general relativity, which explains the large-scale properties of the universe, with quantum mechanics, which explains the behaviour of matter at the atomic and subatomic level. As currently formulated, these two theories are incompatible. String theory may be able to solve this incompatibility issue, which is why it is sometimes described as the "theory of everything" and has generated so much excitement among physicists.
"The difficulty with string theory," says Dr. Jeffrey, "is that its predictions are outside testing with current technology, but it's resolved a lot of theoretical physics puzzles. It's the first theory to reconcile general relativity and quantum theory. String theory has spawned a great deal of definitely useful mathematics." One example is that string theory leads to a new way of proving the basic facts of Morse theory, a sub-discipline of topology or surfaces.
Dr. Jeffrey has been doing her part to develop "definitely useful mathematics" during a year-long stint at the Fields Institute for Research in Mathematical Sciences in Toronto. This residency was part of her 2004 Steacie Fellowship from Science and Engineering Research Canada. It's the latest in a list of awards, including the 2002 Coxeter-James Lectureship and the 2001 Krieger-Nelson Prize from the Canadian Mathematical Society (for outstanding research by a young mathematician and by a woman mathematician, respectively), the University of Toronto's 2000 McLean Award, and the 1999 Premier's Research Excellence Award given by the Ontario government.
Marrying physics and math
For people working in string theory, physicist Ed Witten's mathematical imagination is well-known, and to be able to ground his ideas with solid proofs is no small feat. That's one of the strengths of symplectic geometry: it's a very useful marriage of physics and math.
Although symplectic geometry is a hotbed of activity these days, it's not a new field of study. Its roots can be traced back to 19th-century mathematicians and physicists who were trying to advance the understanding of new ideas in physics. Their work fueled the development of a new kind of mathematics to describe the physics. The word "symplectic" is derived from the Greek "together plaited." It's a fitting description of the powerful mathematical tool created by weaving together geometry and classical mechanics.
In layman's terms, the strength of symplectic geometry is that it preserves area (in two dimensions) no matter what the change in topology or surface shape. This provides a convenient means of dealing with string theory's rolled up, doughnut-shaped dimensions and quantum field theory's infinite dimensions.
Dr. Jeffrey has a foot in both the physics and math camps. She grew up in Toronto where her stepfather Alan Weatherley, a U of T professor emeritus, taught zoology. She did her undergraduate degree in physics at Princeton but then switched to math. When asked why, she shrugs and smiles. "I just was more interested in mathematics than physics."
She followed her physics degree with a master's in mathematics at Cambridge University and a doctorate at Oxford with Sir Michael Atiyah, a world-renowned mathematician. Dr. Jeffrey is one of many young women who've gravitated to symplectic geometry, and she credits Dr. Atiyah for being a supportive supervisor. Two other women pursuing careers in advanced mathematics - Frances Kirwan, now a professor at Oxford, and math prodigy Ruth Lawrence, now at Hebrew University - also did their doctoral degrees with him a few years earlier.
"I was neither the first nor the best," Dr. Jeffrey says modestly with a quick grin. "That took the pressure off of me."
Dusa McDuff, Distinguished Professor of Mathematics at SUNY Stony Brook in New York and one of Dr. Jeffrey's colleagues, says Dr. Atiyah influenced her own early years in symplectic geometry as well, albeit in a more behind-the-scenes fashion. But she thinks his impact on the number of women in the field has more to do with the fact that he mentored two - Drs. Jeffrey and Kirwan - who've gone on to become leaders in symplectic geometry and who, in turn, have attracted other women to the field.
"I attribute it to a combination of circumstances," says Dr. McDuff of the significant number of women in the field. "It used to be a new field, which makes attitudes less entrenched, and it may seem easier to get a foothold because problems are more accessible. It certainly may be encouraging to young women entering the field to see established female mathematicians in it."
Dr. Jeffrey agrees that having women in the field has attracted more women.
"We've nucleated. I am aware of at least one woman - one of my postdocs - who said it was the presence of women in the field and the supportive environment for women that attracted her to the area. Our department has a strong symplectic geometry group with two women and two men and five or six graduate students." The grad students are currently all men but she calls that "an anomoly" - three postdocs last year were women, as was her summer master's student and two PhDs who completed in 2003 and 2004. "And we do attempt to provide support for younger women, such as [offering] a series of lunches in the department of mathematics geared to women."
She recalls helping organize a conference on symplectic geometry about 10 years ago, where many of the speakers were women. "Some of the people attending lectures out of general interest asked if this had been deliberate. It wasn't. We invited them because they were the natural people to choose, the leaders in the field."
Dr. Jeffrey, who has concluded her work at the Fields Institute, says that many interesting problems are coming out of the string theory program. These are problems she's looking forward to tackling.
"My research uses techniques from pure mathematics to prove results obtained by theoretical physicists. Physicists have discovered many formulas that are very surprising and unexpected to pure mathematicians. These formulas can often be proved by orthodox mathematical methods, although pure mathematicians might have been unlikely to have ever come up with them without recourse to the ideas of physicists."
It's fun to be on the cutting edge of a discipline that is showing great promise for the application of symplectic geometry to a range of physics problems. "In the 19th century," she says, leaning forward intently to make her point, "math changed in response to questions in physics. It's really exciting that this has resumed. Math is evolving in response to physics, and this time it's string theory."